# outer measure

Definition [1, 2, 3] Let $X$ be a set, and let $\mathcal{P}(X)$ be the power set of $X$. An outer measure on $X$ is a function $\mu^{\ast}:\mathcal{P}(X)\to[0,\infty]$ satisfying the properties

1. 1.

$\mu^{\ast}(\emptyset)=0$.

2. 2.

If $A\subset B$ are subsets in $X$, then $\mu^{\ast}(A)\leq\mu^{\ast}(B)$.

3. 3.

If $\{A_{i}\}$ is a countable collection of subsets of $X$, then

 $\mu^{\ast}(\bigcup_{i}A_{i})\leq\sum_{i}\mu^{\ast}(A_{i}).$

Here, we can make two remarks. First, from (1) and (2), it follows that $\mu^{\ast}$ is a positive function on $\mathcal{P}(X)$. Second, property (3) also holds for any finite collection of subsets since we can always append an infinite sequence of empty sets to such a collection.

## References

• 1 A. Mukherjea, K. Pothoven, Real and Functional analysis, Plenum press, 1978.
• 2 A. Friedman, Foundations of Modern Analysis, Dover publications, 1982.
• 3 G.B. Folland, Real Analysis: Modern Techniques and Their Applications, 2nd ed, John Wiley & Sons, Inc., 1999.
Title outer measure OuterMeasure 2013-03-22 13:45:20 2013-03-22 13:45:20 mathcam (2727) mathcam (2727) 6 mathcam (2727) Definition msc 60A10 msc 28A10 CaratheodorysExtensionTheorem CaratheodorysLemma ProofOfCaratheodorysExtensionTheorem